8. A particle has a speed given by v = c + bt 3 , what would be the displacement of the particleafter 3 seconds have gone by? What would be its acceleration at 3 seconds?
Question
- A particle has a speed given by v = c + bt 3 , what would be the displacement of the particleafter 3 seconds have gone by? What would be its acceleration at 3 seconds?
Solution
To find the displacement of the particle after 3 seconds, we need to integrate the velocity function with respect to time.
Given that the velocity function is v = c + bt^3, we can integrate it to find the displacement function.
∫(v) dt = ∫(c + bt^3) dt
Integrating c with respect to t gives ct, and integrating bt^3 with respect to t gives (b/4)t^4.
So, the displacement function is given by s = ct + (b/4)t^4.
To find the displacement after 3 seconds, we substitute t = 3 into the displacement function.
s = c(3) + (b/4)(3)^4
s = 3c + (b/4)(81)
Now, to find the acceleration at 3 seconds, we need to differentiate the velocity function with respect to time.
Given that the velocity function is v = c + bt^3, we can differentiate it to find the acceleration function.
dv/dt = d(c + bt^3)/dt
Differentiating c with respect to t gives 0, and differentiating bt^3 with respect to t gives 3bt^2.
So, the acceleration function is given by a = 3bt^2.
To find the acceleration at 3 seconds, we substitute t = 3 into the acceleration function.
a = 3b(3)^2
a = 27b.
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